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On Moment Sequences and Mixed Poisson Distributions
ON MOMENT SEQUENCES AND MIXED POISSON DISTRIBUTIONS MARKUS KUBA AND ALOIS PANHOLZER Abstract. In this article we survey properties of mixed Poisson dis- tributions and probabilistic aspects of the Stirling transform: given a non-negative random variable X with moment sequence (µs)s∈N we de- termine a discrete random variable Y , whose moment sequence is given by the Stirling transform of the sequence (µs)s∈N, and identify the dis- tribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a simple limit theorem based on ex- pansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn mod- els, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time n. We discuss the branch- ing structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley-trees, a parameter in parking functions, zero con- tacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics. 1. Introduction In combinatorics the Stirling transform of a given sequence (as)s∈N, see [12, 75], is the sequence (bs)s∈N, with elements given by ( ) s s b = X a , s ≥ 1. -
Arxiv:1311.5067V5 [Math.CO] 27 Jan 2021 and Φ Eso H Rtadscn Id Ohplnma Aiisapa As Appear Su Families Order
Multivariate Stirling Polynomials of the First and Second Kind Alfred Schreiber Department of Mathematics and Mathematical Education, University of Flensburg, Auf dem Campus 1, D-24943 Flensburg, Germany Abstract Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polyno- −(2n−1) mials Bn,k and a new family Sn,k ∈ Z[X1, . ,Xn−k+1] such that X1 Sn,k and Bn,k obey an inversion law which generalizes that of the Stirling num- bers of the first and second kind. Both polynomial families appear as Lie coefficients in expansions of certain derivatives of higher order. Substituting Dj(ϕ) (the j-th derivative of a fixed function ϕ) in place of the indeterminates Xj shows that both Sn,k and Bn,k are differential polynomials depending on ϕ and on its inverse ϕ, respectively. Some new light is shed thereby on Comtet’s solution of the Lagrange inversion problem in terms of the Bell polynomials. According to Haiman and Schmitt that solution is essentially the antipode on the Fa`adi Bruno Hopf algebra. It can be represented by −(2n−1) X1 Sn,1. Moreover, a general expansion formula that holds for the whole family Sn,k (1 ≤ k ≤ n) is established together with a closed expression for the coefficients of Sn,k. Several important properties of the Stirling numbers are demonstrated to be special cases of relations between the corresponding arXiv:1311.5067v5 [math.CO] 27 Jan 2021 polynomials. As a non-trivial example, a Schl¨omilch-type formula is derived expressing Sn,k in terms of the Bell polynomials Bn,k, and vice versa. -
An Identity for Generalized Bernoulli Polynomials
1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.11.2 47 6 23 11 An Identity for Generalized Bernoulli Polynomials Redha Chellal1 and Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria [email protected] [email protected] [email protected] Mohamed Mehbali Centre for Research Informed Teaching London South Bank University London United Kingdom [email protected] Abstract Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli poly- nomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identi- ties for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling. 1Corresponding author. 1 1 Introduction Let N and C denote, respectively, the set of positive integers and the set of complex numbers. (α) In his book, Roman [41, p. 93] defined generalized Bernoulli polynomials Bn (x) as follows: for all n ∈ N and α ∈ C, we have ∞ tn t α B(α)(x) = etx. (1) n n! et − 1 Xn=0 The Bernoulli numbers Bn, classical Bernoulli polynomials Bn(x), and generalized Bernoulli (α) numbers Bn are, respectively, defined by (1) (α) (α) Bn = Bn(0), Bn(x)= Bn (x), and Bn = Bn (0). (2) The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory, mathematical analysis, and topology. -
Expansions of Generalized Euler's Constants Into the Series Of
Journal of Number Theory 158 (2016) 365–396 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt Expansions of generalized Euler’s constants into −2 the series of polynomials in π and into the formal enveloping series with rational coefficients only Iaroslav V. Blagouchine 1 University of Toulon, France a r t i c l e i n f o a b s t r a c t Article history: In this work, two new series expansions for generalized Received 1 January 2015 Euler’s constants (Stieltjes constants) γm are obtained. The Received in revised form 26 June first expansion involves Stirling numbers of the first kind, 2015 − contains polynomials in π 2 with rational coefficients and Accepted 29 June 2015 converges slightly better than Euler’s series n−2. The Available online 18 August 2015 Communicated by David Goss second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and Keywords: involves Bernoulli numbers with a non-linear combination of Generalized Euler’s constants generalized harmonic numbers. It also permits to derive an Stieltjes constants interesting estimation for generalized Euler’s constants, which Stirling numbers is more accurate than several well-known estimations. Finally, Factorial coefficients in Appendix A, the reader will also find two simple integral Series expansion definitions for the Stirling numbers of the first kind, as well Divergent series Semi-convergent series an upper bound for them. Formal series © 2015 Elsevier Inc. All rights reserved. Enveloping series Asymptotic expansions Approximations Bernoulli numbers Harmonic numbers Rational coefficients Inverse pi E-mail address: [email protected]. -
Multivariate Expansions Associated with Sheffer-Type Polynomials and Operators
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 1 (2006), No. 4, pp. 451-473 MULTIVARIATE EXPANSIONS ASSOCIATED WITH SHEFFER-TYPE POLYNOMIALS AND OPERATORS BY TIAN XIAO HE, LEETSCH C. HSU, AND PETER J.-S. SHIUE Abstract With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general ex- pansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solu- [ tion to the problem of the expansion of A(tˆ)f(g(t)) (a composi- tion of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory. 1. Introduction The purpose of this paper is to study the following expansion problem. Problem 1. Let tˆ = (t1,t2,...,tr), A(tˆ), g(t) = (g1(t1), g2(t2),..., gr(tr)) and f(tˆ) be any given formal power series over the complex number Cr ˆ ′ d field with A(0) = 1, gi(0) = 0 and gi(0) 6= 0 (i = 1, 2, . , r). We wish to find the power series expansion in tˆ of the composite function A(tˆ)f(g(t)). Received October 18, 2005 and in revised form July 5, 2006. d Communicated by Xuding Zhu. AMS Subject Classification: 05A15, 11B73, 11B83, 13F25, 41A58. Key words and phrases: Multivariate formal power series, multivariate Sheffer-type polynomials, multivariate Sheffer-type differential operators, multivariate weighted Stirling numbers, multivariate Riordan array pair, multivariate exponential polynomials. -
Degenerate Stirling Polynomials of the Second Kind and Some Applications
S S symmetry Article Degenerate Stirling Polynomials of the Second Kind and Some Applications Taekyun Kim 1,*, Dae San Kim 2,*, Han Young Kim 1 and Jongkyum Kwon 3,* 1 Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea 2 Department of Mathematics, Sogang University, Seoul 121-742, Korea 3 Department of Mathematics Education and ERI, Gyeongsang National University, Jinju 52828, Korea * Correspondence: [email protected] (T.K.); [email protected] (D.S.K.); [email protected] (J.K.) Received: 20 July 2019; Accepted: 13 August 2019; Published: 14 August 2019 Abstract: Recently, the degenerate l-Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate l-Stirling polynomials as well as the r-truncated degenerate l-Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate l-Stirling polynomials of the second and the r-truncated degenerate l-Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. Keywords: degenerate l-Stirling polynomials of the second kind; r-truncated degenerate l-Stirling polynomials of the second kind; probability distribution 1. Introduction For n ≥ 0, the Stirling numbers of the second kind are defined as (see [1–26]) n n x = ∑ S2(n, k)(x)k, (1) k=0 where (x)0 = 1, (x)n = x(x − 1) ··· (x − n + 1), (n ≥ 1). -
A Determinant Expression for the Generalized Bessel Polynomials
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 242815, 6 pages http://dx.doi.org/10.1155/2013/242815 Research Article A Determinant Expression for the Generalized Bessel Polynomials Sheng-liang Yang and Sai-nan Zheng Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China Correspondence should be addressed to Sheng-liang Yang; [email protected] Received 5 June 2013; Accepted 25 July 2013 Academic Editor: Carla Roque Copyright © 2013 S.-l. Yang and S.-n. Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers. − 1. Introduction Let (, ) denote the coefficient of in −1().We call (, ) a signless Bessel number of the first kind and − The Bessel polynomials form a set of orthogonal polyno- (, ) = (−1) (, ) aBesselnumberofthefirst mials on the unit circle in the complex plane. They are kind. The Bessel number of the second kind (, ) is important in certain problems of mathematical physics; defined to be the number of set partitions of [] := for example, they arise in the study of electrical net- {1,2,3,...,} into blocks of size one or two. Han and works and when the wave equation is considered in spher- Seo [3] showed that the two kinds of Bessel numbers are ical coordinates. -
A Note on Some Identities of New Type Degenerate Bell Polynomials
mathematics Article A Note on Some Identities of New Type Degenerate Bell Polynomials Taekyun Kim 1,2,*, Dae San Kim 3,*, Hyunseok Lee 2 and Jongkyum Kwon 4,* 1 School of Science, Xi’an Technological University, Xi’an 710021, China 2 Department of Mathematics, Kwangwoon University, Seoul 01897, Korea; [email protected] 3 Department of Mathematics, Sogang University, Seoul 04107, Korea 4 Department of Mathematics Education and ERI, Gyeongsang National University, Gyeongsangnamdo 52828, Korea * Correspondence: [email protected] (T.K.); [email protected] (D.S.K.); [email protected] (J.K.) Received: 24 October 2019; Accepted: 7 November 2019; Published: 11 November 2019 Abstract: Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind. Keywords: Bell polynomials; partially degenerate Bell polynomials; new type degenerate Bell polynomials MSC: 05A19; 11B73; 11B83 1. Introduction Studies on degenerate versions of some special polynomials can be traced back at least as early as the paper by Carlitz [1] on degenerate Bernoulli and degenerate Euler polynomials and numbers. In recent years, many mathematicians have drawn their attention in investigating various degenerate versions of quite a few special polynomials and numbers and discovered some interesting results on them [2–9]. -
René Gy Rene. Gy@ Numericable
#A67 INTEGERS 20 (2020) BERNOULLI-STIRLING NUMBERS Ren´eGy [email protected] Received: 5/29/19, Revised: 2/29/20, Accepted: 8/13/20, Published: 8/31/20 Abstract Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious tri- angular array of numbers indexed with positive integers n; k, involving the Bernoulli and cycle Stirling numbers. These numbers are all integers and they vanish when n − k is odd. This triangle has many similarities with the Stirling triangle. In particular, we show how it can be extended to negative indices and how this ex- tension produces a second kind of such integers which may be considered as a new generalization of the Genocchi numbers and for which a generating function is easily obtained. But our knowledge of these integers remains limited, especially for those of the first kind. 1. Introduction Let n and k be non-negative integers and let the generalized Harmonic numbers (k) (k) Hn , Gn be defined as n X 1 H(k) := ; n jk j=1 (k) X 1 Gn := , i1i2 ··· ik 1≤i1<i2<··<ik≤n (1) (1) Pn 1 (0) (0) with Hn = Gn = j=1 j = Hn = Gn; Hn = n and Gn = 1. We have [10] n + 1 = n!G(k); k + 1 n n k being the cycle Stirling number (or unsigned Stirling number of the first kind), so that the Harmonic and cycle Stirling numbers are inter-related by the convolution k−1 n + 1 X n + 1 k = − (−1)k−jH(k−j) , (1.1) k + 1 n j + 1 j=0 2 which is obtained as a direct application of the well-known relation between ele- mentary symmetric polynomials and power sums [9]. -
Arxiv:2103.13478V2 [Math.NT] 11 May 2021 Solution of the Differential Equation Y = E , Special Values of Bell Polynomials
Solution of the Differential Equation y(k) = eay, Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients Ronald Orozco L´opez Departamento de Matem´aticas Universidad de los Andes Bogot´a, 111711 Colombia [email protected] Abstract In this paper special values of Bell polynomials are given by using the power series solution of the equation y(k) = eay. In addition, complete and partial exponential autonomous functions, exponential autonomous polyno- mials, autonomous polynomials and (k, a)-autonomous coefficients are de- fined. Finally, we show the relationship between various numbers counting combinatorial objects and binomial coefficients, Stirling numbers of second kind and autonomous coefficients. 1 Introduction It is a known fact that Bell polynomials are closely related to derivatives of composi- tion of functions. For example, Faa di Bruno [5], Foissy [6], and Riordan [10] showed that Bell polynomials are a very useful tool in mathematics to represent the n-th derivative of the composition of functions. Also, Bernardini and Ricci [2], Yildiz et arXiv:2103.13478v2 [math.NT] 11 May 2021 al. [12], Caley [3], and Wang [13] showed the relationship between Bell polynomials and differential equations. On the other hand, Orozco [9] studied the convergence of the analytic solution of the autonomous differential equation y(k) = f(y) by using Faa di Bruno’s formula. We can then look at differential equations as a source for investigating special values of Bell polynomials. In this paper we will focus on finding special values of Bell polynomials when the vector field f(x) of the autonomous differential equation y(k) = f(y) is the exponential function. -
Generalized Bell Polynomials and the Combinatorics of Poisson Central Moments
Generalized Bell polynomials and the combinatorics of Poisson central moments Nicolas Privault∗ Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University SPMS-MAS-05-43, 21 Nanyang Link Singapore 637371 February 25, 2011 Abstract We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson dis- tribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2. Key words: Bell polynomials, Poisson distribution, central moments, Stirling num- bers. Mathematics Subject Classification: 11B73, 60E07. 1 Introduction The moments of the Poisson distribution are well-known to be connected to the com- binatorics of the Stirling and Bell numbers. In particular the Bell polynomials Bn(λ) satisfy the relation n Bn(λ) = Eλ[Z ]; n 2 IN; (1.1) where Z is a Poisson random variable with parameter λ > 0, and n X Bn(1) = S(n; c) (1.2) c=0 ∗[email protected] 1 is the Bell number of order n, i.e. the number of partitions of a set of n elements. In this paper we study the central moments of the Poisson distribution, and we show that they can be expressed using the number of partitions of a set into subsets of size at least 2, in connection with an extension of the Bell polynomials. Consider the above mentioned Bell (or Touchard) polynomials Bn(λ) defined by the exponential generating function 1 n t X t eλ(e −1) = B (λ); (1.3) n! n n=0 λ, t 2 IR, cf. -
Contents What These Numbers Count Triangle Scheme for Calculations
From Wikipedia, the free encyclopedia In combinatorial mathematics, the Bell numbers count the number of partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. Starting with B0 = B1 = 1, the first few Bell numbers are: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, ... (sequence A000110 in the OEIS). The nth of these numbers, Bn, counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. Outside of mathematics, the same number also counts the number of different rhyme schemes for n-line poems.[1] As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, Bn is the nth moment of a Poisson distribution with mean 1. Contents 1 What these numbers count 1.1 Set partitions 1.2 Factorizations 1.3 Rhyme schemes 1.4 Permutations 2 Triangle scheme for calculations 3 Properties 3.1 Summation formulas 3.2 Generating function 3.3 Moments of probability distributions 3.4 Modular arithmetic 3.5 Integral representation 3.6 Log-concavity 3.7 Growth rate 4 Bell primes 5History 6 See also 7Notes 8 References 9 External links What these numbers count Set partitions In general, Bn is the number of partitions of a set of size n.